I've found that the module's hook_block_view implementation won't ever produce a match on $delta if multiple blocks are added.

Current code from twitter_pull.module:

<?php
/**
 * Implementation of hook_block_view()
 */
function twitter_pull_block_view($delta = '') {
  $info = twitter_pull_block_data();
  $b_info = $info["$delta"]; 
  $content = twitter_pull_render($b_info->tweetkey, $b_info->title, $b_info->number_of_items, $b_info->theme_key);
  return array("subject"=>"", "content"=>$content);
}
?>
<?php
$b_info = $info["$delta"]; // <- This will never actually match a delta value if the example hook_block format on the project page is followed.
?>

This code works for me:

<?php
/**
 * Implementation of hook_block_view()
 */
function twitter_pull_block_view($delta = '') {
  $info = twitter_pull_block_data();
    
  foreach($info as $key => $value) if($value->delta==$delta) $b_info = $value;
  if(!isset($b_info)) $b_info = $info[0];

  $content = twitter_pull_render($b_info->tweetkey, $b_info->title, $b_info->number_of_items, $b_info->theme_key);
  return array("subject"=>"", "content"=>$content);
}
?>